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On the transport limit of singularly perturbed convection–diffusion problems on networks

Egger, Herbert ; Philippi, Nora (2024)
On the transport limit of singularly perturbed convection–diffusion problems on networks.
In: Mathematical Methods in the Applied Sciences, 2021, 44 (6)
doi: 10.26083/tuprints-00017804
Article, Secondary publication, Publisher's Version

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Item Type: Article
Type of entry: Secondary publication
Title: On the transport limit of singularly perturbed convection–diffusion problems on networks
Language: English
Date: 12 February 2024
Place of Publication: Darmstadt
Year of primary publication: 2021
Place of primary publication: Chichester
Publisher: John Wiley & Sons
Journal or Publication Title: Mathematical Methods in the Applied Sciences
Volume of the journal: 44
Issue Number: 6
DOI: 10.26083/tuprints-00017804
Corresponding Links:
Origin: Secondary publication DeepGreen

We consider singularly perturbed convection–diffusion equations on one‐dimensional networks (metric graphs) as well as the transport problems arising in the vanishing diffusion limit. Suitable coupling conditions at inner vertices are derived that guarantee conservation of mass and dissipation of a mathematical energy which allows us to prove stability and well‐posedness. For single intervals and appropriately specified initial conditions, it is well‐known that the solutions of the convection–diffusion problem converge to that of the transport problem with order O(ϵ) in the L∞(L²)‐norm with diffusion ϵ → 0. In this paper, we prove a corresponding result for problems on one‐dimensional networks. The main difficulty in the analysis is that the number and type of coupling conditions changes in the singular limit which gives rise to additional boundary layers at the interior vertices of the network. Since the values of the solution at these network junctions are not known a priori, the asymptotic analysis requires a delicate choice of boundary layer functions that allows to handle these interior layers.

Uncontrolled Keywords: asymptotic analysis, diffusion and convection (76R05), partial differential equations on networks, singular perturbations in the context of PDEs (35B25)
Status: Publisher's Version
URN: urn:nbn:de:tuda-tuprints-178044
Additional Information:

MSC CLASSIFICATION: 35B25; 35K20; 35R02; 76M45

Classification DDC: 500 Science and mathematics > 510 Mathematics
Divisions: 04 Department of Mathematics > Numerical Analysis and Scientific Computing
Date Deposited: 12 Feb 2024 13:46
Last Modified: 12 Feb 2024 13:46
SWORD Depositor: Deep Green
URI: https://tuprints.ulb.tu-darmstadt.de/id/eprint/17804
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